THE RIEMANN PROBLEM FOR A TWO-DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS WITH NON-CLASSICAL WEAK SOLUTIONS:SOME ONE-J AND NON-R INITIAL DATA

1IntroductionWeareconcernedwiththefollowing2x2nonstrictly11yperbolicsystemofconservationlawswithinitialdatawhere(m,yi),i=1,2,3,areconstantstates.Wecall(1.1),(1.2)aRiemanll-typeproblelll,.Byaweaksolutionto(1.1),(1.2)wemeanapairofgeneralizedfullctions('IL.l,)(x,y.t)satisfyinglffeceivedFeb.17,1995;revisedApr.23,1996.SuPPortedbyNSFC(YOuth).foreacllof(T,y,f)6C:(R'xR )alld(u,U)(:r,;(/,f)~(it..())(x,y.0)ast-- 0 forallllostevery(x,if/,t)ERZxR .111(1.3).wealwaysassulllethat('(ti))(T,y.f)isals…     

A FINITE DIFFERENCE SCHEME FOR THE GENERALIZED NONLINEAR SCHRDINGER EQUATION WITH VARIABLE COEFFICIENTS

1. Generalized Nonlinear Schrsdinger EquationThe Schr6dinger equation has been extensively used in physics research, particularlyin the modeling of nonlinear dispersion waves [8]. Numerical methods for solving theSchr6dinger equation have been discussed in the literature. In this article, we considera generalized nonlinear Schr6dinger equation with variable coefficientsi: ~ g(A(x)Z) iF(t)u B(x) lulp~' u = 0, iZ ~ ~l, P > 1, (1)where u(x, 0) ~ of (x). The coefficients A(x), F(t) and, …     

THE GENERALIZED RIEMANN PROBLEM FOR A SCALAR COMBUSTION MODEL --THE PERTURBATION ON INITIAL BINDING ENERGY

In this paper, we study the generalized Riemann problem for a scalar Chapman-Jouguet combustion model in a neighborhood of the origin on upper half of the （x, t）plane. We focus our attention on the perturbation on initial binding energy. Under the entropy conditions, the solutions are obtained constructively. It shows that the perturbed Riemann solutions possess the structural stability except the case that the corresponding Riemann solutions contain CJDT, for which CJDT may transform into SDT after perturbation on initial binding energy in the neighborhood of the origin.     

The perturbation on initial binding energy for a Majda-CJ combustion model

We investigate the perturbed Riemann problem for a scalar Chapman–Jouguet combustion model – the perturbation on initial binding energy. Under the entropy conditions, we obtain the unique solutions in a neighbourhood of the origin (t?>?0) on the (x,?t) plane. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. That is, the perturbation may transform a Chapman–Jouguet detonation into a strong detonation or a weak deflagration following a shock wave; a strong detonation into a weak deflagration following a shock wave; a Chapman–Jouguet deflagration into a weak deflagration.     

DEPENDENCE OF QUALITATIVE BEHAVIOR OF THE NUMERICAL SOLUTIONS ON THE IGNITION TEMPERATURE FOR A COMBUSTION MODEL

We study the dependence of qualitative behavior of the numerical solutions （obtained by a projective and upwind finite difference scheme） on the ignition temperature for a combustion model problem with general initial condition. Convergence to weak solution is proved under the Courant-Friedrichs-Lewy condition. Some condition on the ignition temperature is given to guarantee the solution containing a strong detonation wave or a weak detonation wave. Finally, we give some numerical examples which show that a strong detonation wave can be transformed to a weak detonation wave under some well-chosen ignition temperature.     

EXISTENCE OF GLOBAL SMOOTH SOLUTION TO INITIAL BOUNDARY VALUE PROBLEM FOR A VISCOELASTIC MODEL WITH RELAXATION

This paper is concerned with the initial boundary value problem for a vis-coelastic model with relaxation. Under the only assumption that the C^0-norm of theinitial data is small, without smallness hypothesis for the C^1-norm, the existence of theglobal smooth solution to the corresponding initial boundary value problem is proved.The analysis is based on some a priori estimates obtained by the “maximum principle” offirst-order quasilinear hyperbolic system.     

BLOW-UP OF THE SOLUTION FOR A CLASS OF POROUS MEDIUM EQUATION WITH POSITIVE INITIAL ENERGY

This paper deals with a class of porous medium equation ut=△u^m＋f（u）with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f（u）.